Question: We are given that $\dfrac{dy}{dx}=\dfrac xy$. Find an expression for $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. $\dfrac{d^2y}{dx^2}=$
Notice that the equation defines $y$ implicitly—we don't have an explicit expression for $y$ in terms of $x$. So we will have to use implicit differentiation. If we differentiate the equation once, we will have $\dfrac{d^2y}{dx^2}$ on the left-hand side of the equation. This is what we get after we differentiate each side of the equation: $\dfrac{d^2y}{dx^2}=\dfrac{y-\dfrac{dy}{dx}\cdot x}{y^2}$ The expression we got for $\dfrac{d^2y}{dx^2}$ isn't in terms of only $x$ and $y$ because it includes $\dfrac{dy}{dx}$ in it. Let's plug the original equation ${\dfrac{dy}{dx}=\dfrac xy}$ in the new equation to obtain an expression in terms of only $x$ and $y$ : $\begin{aligned} \dfrac{d^2y}{dx^2}&=\dfrac{y-{\dfrac{dy}{dx}}\cdot x}{y^2} \\\\ &=\dfrac{y-{\dfrac xy}\cdot x}{y^2} \\\\ &=\dfrac{y^2-x^2}{y^3} \end{aligned}$ In conclusion, this is an expression for $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$ : $\dfrac{d^2y}{dx^2}=\dfrac{y^2-x^2}{y^3}$